German Patent Application No. DE 10 2004 041 767 A1 describes a method and a device for operating an internal combustion engine having an exhaust gas recirculation system that enables a diagnosis of an exhaust gas recirculation cooling device during normal operation of the internal combustion engine. Here, a characteristic quantity for the functioning of the exhaust gas recirculation cooling device is monitored. The characteristic quantity for the function of the exhaust gas recirculation cooling device is determined as a function of a measurement value. The characteristic quantity for the functioning of the exhaust gas recirculation cooling device is prespecified assuming an intact exhaust gas recirculation cooling device. The determined value for the characteristic quantity for the functioning of the exhaust gas recirculation cooling device is compared to the prespecified value. If the determined value of the characteristic quantity for the functioning of the exhaust gas recirculation cooling device deviates from the prespecified value, an error is recognized. In addition, a bypass around the exhaust gas recirculation cooling device is provided with a bypass valve. When the bypass valve is open, the recirculated exhaust gas is conducted at least partly through the bypass. When the bypass valve is closed, the recirculated exhaust gas is conducted through the exhaust gas recirculation cooling device.